Anthony Binford's Godfather was "ANTHONY JONES" not "ANTHONY SMITH".
Genealogies of Virginia Families From Tyler's Quarterly........ Volume II Gildart - Pettus
IN THE NAME OF GOD, AMEN, - the 16th day of August, 1649. I Anthony Jones of the Isle of Wright of Virginia........ ........ITEM, I bequeath to my Godson Anthony Benford (Binford) one Heifer to be paid the next May.
I don't believe the Binford's were Quakers at this time but members of the Church of England.